The present invention relates to filters such as the kind that can be utilized in communications systems. FIG. 1 illustrates a basic block diagram of a digital communication system that utilizes pulse amplitude modulation. In this system, a pulse generator 12 receives clock pulses and binary input data. The output of pulse generator 12 will be a digital binary stream of pulses.
The pulse stream from pulse generator 12 is applied to the digital transmitting filter 14 that shapes the pulse for output to the digital-to-analog converter 15 and transmission over channel 16. Channel 16 may be a wired or wireless channel depending upon the application. The transmitted data is received at receiving filter 18. The output of filter 18 is applied to analog-to-digital converter 20. Analog-to-digital converter 20 utilizes clock pulses that are generally recovered from the transmitted data by clock recovery circuit 22. The output binary data from analog-to-digital converter 20 is a replica of the input binary stream that was provided to pulse generator 12.
Major objectives of the design of the baseband PAM system are to choose the transmitting and receiving filters 14 and 18 to minimize the effects of noise, to eliminate or minimize inter-symbol interference (ISI) and to reduce stop band energy. Inter-symbol interference can be theoretically eliminated by properly shaping the pulses of the transmitted signal. This pulse shaping can be accomplished by causing the pulse to have a zero value at periodic intervals.
Modern embodiments of pulse shaping filters use a pair of matched filters, one for transmit and one for receive. The convolution of the transmit filter with the receive filter forms the complete pulse shaping filter. Inter-symbol interference is avoided since the combined filter impulse response reaches unity at a single point and is zero periodically at every other information point (Nyquist sampling rate). The linear superposition of pulses representing a pulse train preserves bandwidth and information content. Linear superposition of band limited pulses remains band limited and sampling the combined filter at the information rate recovers the information.
FIG. 3b shows an example of a Nyquist filter impulse response. Zeros occur at the information rate, except at one information bearing point. All Nyquist filters having the same stop band are equally bandwidth limited if the time response of the filters is allowed to go to infinity. Realizable filters, however, are truncated in time since it is not possible to have an infinitely long time function. Truncation error in the time domain causes the theoretical stop band achievable by all Nyquist filters to be violated, so that out of band energy exists in excess of the stop band frequency.
The most bandwidth efficient filter is the “brick wall” filter illustrated in FIG. 3a by the box (α=0). The time response of this filter is shown in FIG. 3b (α=0). While bandwidth efficiency is theoretically greatest for a brick wall filter as the time response approaches infinity, truncation error causes poor performance for practical and realizable approximations to the brick wall filter.
One method of producing practical filters is to allow the stop band of Nyquist compliant filters to exceed the bandwidth of the ideal brick wall filter and smoothly transition to the stop band. A class of such filters is the raised cosine filters. In the frequency domain (FIG. 3a), the raised cosine filter smoothly approaches the frequency stop band (except for the limiting brick wall filter case). The raised cosine filter is continuous at the stop band and the first derivative is continuous. The second derivative of a raised cosine filter, however, is not continuous at the stop band.
In current embodiments of most systems, the raised cosine filter is used in its matched filter version. The transmit square root raised cosine filter, which determines the spectral bandwidth efficiency of the system, is discontinuous in the first derivative at the stop band.